Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data

TL;DR

This paper investigates the lifespan of solutions to a semilinear wave equation with time-dependent damping, establishing sharp bounds and thresholds for blowup and global existence based on initial data and critical exponents.

Contribution

It provides a sharp upper bound for the lifespan of solutions with small initial data and clarifies the critical exponent dividing blowup and global existence in three dimensions.

Findings

Sharp upper bounds for solution lifespan in terms of small parameter p.

Identification of the threshold exponent p_0(N+m) for blowup versus global existence.

Construction of novel test functions involving Gauss hypergeometric functions.

Abstract

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+\mu)$, where $p_F(N)$ denotes the Fujita exponent for the nonlinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonlinear wave equation in $n$-dimension with $\mu=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+\mu)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the conjugate linear equation $\pa_t^2\Phi-\Delta \Phi-\pa_t(\frac{\mu}{1+t}\Phi)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.